Optical resonator

1. What a Resonator (Cavity) Is

Optical resonator / cavity = two parallel mirrors ( $M1$ highly reflective $R≈100$ %, $M2$ partially transmitting $R≈95$ %) enclosing the gain medium, providing feedback and mode selection.

Without resonator, stimulated photons leave — no amplification; with resonator, photons bounce many times ⇒ photon density $ρ$ builds up ⇒ gain > loss ⇒ laser oscillation.

2. Conditions for Lasing (gain + resonator)

Threshold condition:

g_{peak} > α_{loss} + (1/2L) ln(1/R_1R_2)

where $g$ is gain coefficient (∝ $N_2-N_1$ ), $α$ internal loss, $L$ cavity length, $R_1,R_2$ mirror reflectivities.

Phase condition: round-trip phase $2kL =2nπ$ ⇒

L = mλ/2

⇒ only longitudinal modes spaced

Δν = c/(2L)

(for linear Fabry-Perot) oscillate within gain linewidth.

Resonator selects one axial direction ⇒ directionality.

3. Types of Cavities

Fabry-Perot (plane-parallel): simple, but unstable if misaligned.
Confocal / concentric / hemispherical: curved mirrors for stability (stability condition $0<g_1g_2<1$ where $g=1-L/R$ ).
Ring cavity: three mirrors in triangle (used in gyros).

Active medium often at mode waist.

4. Functions of Resonator (exam list)

Feedback — returns photons to gain medium for further stimulated emission (amplification loop).
Mode selection — only $ν$ satisfying $2L=mλ$ and within gain bandwidth oscillate ⇒ spectral narrowing (monochromatic).
Direction selection — photons off-axis walk out after few bounces ⇒ only axial photons stay ⇒ collimated beam.
Coherence building — many round trips phase-lock photons ⇒ high coherence.

5. Worked Note

He-Ne cavity L=30 cm: Δν=c/2L=500 MHz; gain linewidth ~1.5 GHz ⇒ ~3 modes can oscillate; etalon selects one.

Ruby cavity L=10 cm: mode spacing c/2L=1.5 GHz; gain linewidth ~300 GHz ⇒ many modes, but Q-switch selects.

Threshold: if R1=100%, R2=95%, L=30 cm, α≈0.01 cm⁻¹: need g> α + ln(1/0.95)/0.6 ≈0.01+0.085≈0.095 cm⁻¹.

6. Exam Points

Define resonator and draw Fabry-Perot cavity — 5 marks, very frequent.
Explain feedback and threshold $g>loss$ — 5 marks, very frequent.
Phase condition $2L=mλ$ , mode spacing $c/2L$ — 5 marks, frequent.
Functions (feedback, mode/direction selection) — 5 marks, very frequent.

Optical resonator